# The CES Production Function

**Posted:**March 5, 2010

**Filed under:**Uncategorized 1 Comment

In many fields of economics, a particular class of functions called *Constant Elasticity of Substitution *(CES) functions are privileged because of their invariant characteristic, namely that the elasticity of substitution between the parameters is constant on their domains (for a definition of elasticity, take a glance at this post).

A standard production function, linking the production factors to output is of the form:

where *Y *is the total production, *L *the quantity of human capital (labour, measured in a unit like *man-hours*) used and *K *the quantity of phyisical capital used (measured in a unit like *machine-hours*). Generally speaking, the production function for* ** *production factors is of the form:

Where is the quantity of factor *i *used. The factors generalized *CES *production function (also called the Armington aggregator) is:

With . First and foremost, to study the returns to scale (refer to the link if you are not familiar with a formal definition of returns to scale) of the function, we shall study its homogeneity.

In other words, the *CES *function is homogeneous of degree . Hence:

- If , the returns to scale are increasing.
- If , the returns to scale are decreasing.
- If , the returns to scale are constant.

The other parameters are: ** **

**Relative weight:**The parameter associated with each production factor represents its relative distributional weight,*i.e.*the significance in the production.

**Elasticity of substitution:**The elasticity of substitution, as the function indicates, is constant. It is: (see the addendum to this post).

What is so interesting about this function is that one can derive special cases from it:

- If , we obtain the
**perfect substitutes production function**: . - If , we obtain the
**Leontief production function**, also known as the perfect complements production function: - If , we obtain the
**Cobb-Douglas production function**, also known as the imperfect complements production function: .

Proving such assertions is far from trivial and my next blog post will be dedicated to deriving the Cobb-Douglas production function from the *CES *function.

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